An analog frequency filter is a common building block in communication, audio, signal processing, and the like. An analog active filter is realized, in a semiconductor integrated circuit, by an operational amplifier, a resistor(s), a capacitor(s), and an inductor(s). The operational amplifier serves as an active element, whereas the resistor(s), the capacitor(s), and the inductor serve as passive components.
In general, integration of inductors in a semiconductor occupies a large area, and is avoided, accordingly. Among the many kinds of active filters, a switched capacitor filter, carrying out a time-sample operation in which the resistors are replaced with switched capacitors, is widely used in analog signal processing applications. The switched-capacitor filter has an advantage because the transfer function of the filter is determined by a capacitance ratio and a sampling frequency. Hence, accurate characteristics can be realized.
For continuous time signal processing, RC active filters, MOSFET-C filters, and Gm-C filters are also used. In these realizations, circuit or filter characteristics vary depending on equivalent RC (resistor-capacitor) products, a resistance ratio, and a capacitance ratio.
A transfer function H(s) of a typical analog filter can be found by the product of a single-pole H1(s) transfer function and a second-order (Biquad) H2(s) transfer function. Considered here is a concrete example of a low-pass filter for implementing a fourth-order Butterworth response. Two Biquad sections connected in cascade can be used to realize the filter as shown in FIG. 4(b).
FIG. 4(a) is a circuit diagram illustrating a circuitry of an analog active RC filter realized as the Biquad section. If transfer functions of first and second stages of the Biquad sections are indicated by H21(s) and H22(s), respectively, then a transfer function H(s) of FIG. 4(b) satisfies the following formula:
                                          H            ⁡                          (              s              )                                =                                                    H                21                            ⁡                              (                s                )                                      ×                                          H                22                            ⁡                              (                s                )                                                    ⁢                                  ⁢                              H            21                    =                                    -                              (                                                      R                    21                                                        R                    11                                                  )                                                                                                          1                    +                                          s                      ⁢                                                                                          ⁢                                              C                        21                                            ⁢                                                                        R                          31                                                ⁡                                                  (                                                      1                            +                                                                                          R                                21                                                                                            R                                11                                                                                      +                                                                                          R                                21                                                                                            R                                31                                                                                                              )                                                                                      +                                                                                                                                          s                      2                                        ⁢                                          C                      11                                        ⁢                                          C                      21                                        ⁢                                          R                      21                                        ⁢                                          R                      31                                                                                                          ⁢                                  ⁢                              H            22                    =                                    -                              (                                                      R                    22                                                        R                    12                                                  )                                                                                                          1                    +                                          s                      ⁢                                                                                          ⁢                                              C                        22                                            ⁢                                                                        R                          32                                                ⁡                                                  (                                                      1                            +                                                                                          R                                22                                                                                            R                                12                                                                                      +                                                                                          R                                22                                                                                            R                                32                                                                                                              )                                                                                      +                                                                                                                                          s                      2                                        ⁢                                          C                      12                                        ⁢                                          C                      22                                        ⁢                                          R                      22                                        ⁢                                          R                      32                                                                                                                              [                  Formula          ⁢                                          ⁢          1                ]            
If it is assumed that each resistance in the formula is equal to R, the formula can be simplified as follows:
                                          H            21                    =                                    -              1                                      1              +                              s                ⁢                                                                  ⁢                3                ⁢                                  C                  21                                ⁢                R                            +                                                s                  2                                ⁢                                  C                  11                                ⁢                                  C                  21                                ⁢                                  R                  2                                                                    ⁢                                  ⁢                              H            22                    =                                    -              1                                      1              +                              s                ⁢                                                                  ⁢                3                ⁢                                  C                  22                                ⁢                R                            +                                                s                  2                                ⁢                                  C                  12                                ⁢                                  C                  22                                ⁢                                  R                  2                                                                                        [                  Formula          ⁢                                          ⁢          2                ]            
C2j is assumed to be reference capacitance of each stage, and a ratio kj of the j-th stage is defined as follows:
                                          H            21                    =                                                                      -                  1                                                  1                  +                                      s                    ⁢                                                                                  ⁢                    3                    ⁢                                          C                      21                                        ⁢                    R                                    +                                                            s                      2                                        ⁢                                          k                      1                                        ⁢                                          C                      21                      2                                        ⁢                                          R                      2                                                                                  ⁢                                                          ⁢                              k                1                                      =                                          C                11                                            C                21                                                    ⁢                                  ⁢                              H            22                    =                                                                      -                  1                                                  1                  +                                      s                    ⁢                                                                                  ⁢                    3                    ⁢                                          C                      22                                        ⁢                    R                                    +                                                            s                      2                                        ⁢                                          k                      2                                        ⁢                                          C                      22                      2                                        ⁢                                          R                      2                                                                                  ⁢                                                          ⁢                              k                2                                      =                                          C                12                                            C                22                                                                        [                  Formula          ⁢                                          ⁢          3                ]            
For example, for the fourth-order low-pass Butterworth response, the following ratios are found:
k1=15.36 and k2=2.636
In general, as the order of the filter increases, the capacitance ratio increases. As another example, for an eighth-order low-pass Butterworth filter, a maximum capacitance ratio satisfies: kmax=59.1. Meanwhile, for the purpose of reducing an area that the capacitor occupies, a capacitor having very small capacitance of approximately 0.2 pF or less is used. However, it is difficult to control the very small capacitance of such a capacitor. Thus, a large capacitance ratio is undesirable. For example, if it is assumed that the maximum capacitance of a capacitor for use in the eighth-order filter is limited to 5 pF (C1), then a capacitance C2 satisfies C2=5 pF/59.1(≈0.084 pF), which is on the order of the layout parasitic capacitance, and therefore difficult to control.
U.S. Pat. No. 4,498,063 (hereinafter, referred to as “patent document 1”, and incorporated by reference) presents a technique to reduce the capacitance needed in switched capacitor filters (see FIG. 5). In the patent document 1, the use of a resistor voltage divider reduces the capacitance ratio.
Specifically, the circuit disclosed in the patent document 1, just uses a resistor voltage divider (R1-R2) which attenuates the input signal. For instance, if the resistor voltage divider is not used, then the voltage gain of the circuit is represented by the following Formula 4, where the input signal is indicated by Vin, an output signal is indicated by Vout, capacitance of a feedback capacitor 19 is indicated by αC. Note that C′4 indicates capacitance C4 in the case where no resistor divider is used.
                                          V            out                                V            in                          =                  -                                    C              4              ′                                      α              ⁢                                                          ⁢              C                                                          [                  Formula          ⁢                                          ⁢          4                ]            
The use of the resistor divider causes the input signal to be attenuated by a factor K, then, the following formula 5 is satisfied.
                                          V            out                                V            in                          =                              -                                          C                4                ′                                            α                ⁢                                                                  ⁢                C                                              ×          K                                    [                  Formula          ⁢                                          ⁢          5                ]            
As such, if a new capacitor C4=C′4/K is used, then the same voltage gain is obtained. However, note in this case that the capacitance C4 becomes larger than the capacitance C′4.
This technique, however, is not adequate because the reduction of the input signal degrades the Signal-to-Noise ratio.
The circuit disclosed in U.S. Pat. No. 4,743,872 (hereinafter, referred to as “patent document 2”, and incorporated by reference), similarly with the patent document 1, is intended for switched capacitor filters, in which three capacitors 3a, 3b, and 3c are arranged so as to have a T-connection (see FIG. 6). This arrangement allows the realization of reduced effective capacitance CQ. A resistor 8 having high resistance values determines a potential at a joint node of the capacitors 3a, 3b, and 3c. 
As shown in FIG. 6, the circuit disclosed in the U.S. Pat. No. 4,743,872, as the U.S. Pat. No. 4,498,063, is also intended for switched capacitor filters, and therefore operates using switches 2a, 2b, 2c and 2d. As a switched capacitor circuit, when the mentioned switches are commutating the capacitors 3a, 3b and 3c behave as an equivalent resistor, rather than a capacitor.
The main drawbacks of this technique are: (i) the high-value resistor introduces thermal noise into the capacitor, and (ii) the clocking of the capacitors requires a sampling frequency much higher than the cut-off frequency of the filter. This causes an increase in switching noise and power dissipation.
The analog frequency filter is a common building block in communication, audio, signal processing, and the like. The state-of-the-art implementations of high-order active R-C filters suffer from the need of high ratio of capacitances. The prior art techniques to reduce the value of capacitance ratio relies on the use of resistor-dividers (e.g., U.S. Pat. No. 4,498,063), which attenuates the input signal. This causes deterioration of the signal-to-noise ratio (SNR), when the prior art techniques are used in wireless transceivers.
Further, the prior art techniques to reduce the value of capacitance ratio uses a switched capacitor network (e.g., U.S. Pat. No. 4,743,872) with the junction node connected to a reference voltage through a high value resistor. As a result, (i) the high-value resistor introduces thermal noise into the capacitor, and (ii) the clocking of the capacitors requires a sampling frequency much higher than the cut-off frequency of the filter. This causes an increase in switching noise and power dissipation.